Citation: Haah, Jeongwan. "Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices." Journal of Mathematical Physics 62.1 (2021).
Web: https://arxiv.org/abs/1812.11193
Tags: Toric-code, MTC-reconstruction, Abelian-anyons
This paper is devoted to proving one main theorem: every frustration-free translation-invariant topological order made of commuting prime-order Pauli stabilizers is equivalent by a finite-depth circuit to a stack of copies of the qudit toric code. Further, this finite-depth circuit can be realized using Clifford gates.
This paper is a lovely result. Fracton codes in 3D can be constructed using frustration-free translation-invariant commuting qubit Pauli stabilizers, and hence this result justifies the fact that exotic examples like that cannot exist in 2D, or at least they require more intricate construction in 2D.
This paper works with the algebra of polynomials and matrices over the finite field with p elements. A key technical ingredient used in the proof is the fact that every phase satisfying the conditions of the theorem has a boson. This is a nontrivial result, which was proved in
> Haah, Jeongwan, Lukasz Fidkowski, and Matthew B. Hastings. "Nontrivial quantum cellular automata in higher dimensions." Communications in Mathematical Physics 398.1 (2023): 469-540.
Note that publication shenanigans have messed up the orders of the dates.